Wavelet transforms for homogeneous mixed-norm Triebel--Lizorkin spaces

TL;DR

This paper extends the classical $$-transform approach to homogeneous mixed-norm Triebel--Lizorkin spaces using wavelet transforms, providing rigorous definitions and properties of the synthesis operator.

Contribution

It introduces a discrete wavelet-based $$-transform for mixed-norm Triebel--Lizorkin spaces and rigorously defines the synthesis operator via Pettis integrals.

Findings

Extended $$-transform to mixed-norm spaces

Rigorous definition of the synthesis operator

Terms can be summed in any order without changing the distribution

Abstract

Homogeneous mixed-norm Triebel--Lizorkin spaces are introduced and studied with the use of a discrete wavelet transformation, the so-called $\varphi$-transform. This extends the classical $\varphi$-transform approach introduced by Frazier and Jawerth to the setting of mixed-norm spaces. Moreover, the theory of the $\varphi$-transform is enhanced through a precise definition of the synthesis operator, in terms of a Pettis integral, and a number of rigorous results for this operator. Especially its terms can always be summed in any order, without changing the resulting distribution.